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## Geometry (all content)

### Course: Geometry (all content)>Unit 13

Lesson 1: Special right triangles

# Special right triangles intro (part 1)

A 45-45-90 triangle is a special type of right triangle, where the ratio of the lengths of the sides of a 45-45-90 triangle is always 1:1:√2, meaning that if one leg is x units long, then the other leg is also x units long, and the hypotenuse is x√2 units long. Created by Sal Khan.

## Video transcript

Welcome to the presentation on 45-45-90 triangles. Let me write that down. How come the pen-- oh, there you go. 45-45-90 triangles. Or we could say 45-45-90 right triangles, but that might be redundant, because we know any angle that has a 90 degree measure in it is a right triangle. And as you can imagine, the 45-45-90, these are actually the degrees of the angles of the triangle. So why are these triangles special? Well, if you saw the last presentation I gave you a little theorem that told you that if two of the base angles of a triangle are equal-- and it's I guess only a base angle if you draw it like this. You could draw it like this, in which case it's maybe not so obviously a base angle, but it would still be true. If these two angles are equal, then the sides that they don't share-- so this side and this side in this example, or this side and this side in this example-- then the two sides are going to be equal. So what's interesting about a 45-45-90 triangle is that it is a right triangle that has this property. And how do we know that it's the only right triangle that has this property? Well, you could imagine a world where I told you that this is a right triangle. This is 90 degrees, so this is the hypotenuse. Right, it's the side opposite the 90 degree angle. And if I were to tell you that these two angles are equal to each other, what do those two angles have to be? Well if we call these two angles x, we know that the angles in a triangle add up to 180. So we'd say x plus x plus-- this is 90-- plus 90 is equal to 180. Or 2x plus 90 is equal to 180. Or 2x is equal to 90. Or x is equal to 45 degrees. So the only right triangle in which the other two angles are equal is a 45-45-90 triangle. So what's interesting about a 45-45-90 triangle? Well other than what I just told you-- let me redraw it. I'll redraw it like this. So we already know this is 90 degrees, this is 45 degrees, this is 45 degrees. And based on what I just told you, we also know that the sides that the 45 degree angles don't share are equal. So this side is equal to this side. And if we're viewing it from a Pythagorean theorem point of view, this tells us that the two sides that are not the hypotenuse are equal. So this is a hypotenuse. So let's call this side A and this side B. We know from the Pythagorean theorem-- let's say the hypotenuse is equal to C-- the Pythagorean theorem tells us that A squared plus B squared is equal to C squared. Right? Well we know that A equals B, because this is a 45-45-90 triangle. So we could substitute A for B or B for A. But let's just substitute B for A. So we could say B squared plus B squared is equal to C squared. Or 2B squared is equal to C squared. Or B squared is equal to C squared over 2. Or B is equal to the square root of C squared over 2. Which is equal to C-- because we just took the square root of the numerator and the square root of the denominator-- C over the square root of 2. And actually, even though this is a presentation on triangles, I'm going to give you a little bit of actually information on something called rationalizing denominators. So this is perfectly correct. We just arrived at B-- and we also know that A equals B-- but that B is equal to C divided by the square root of 2. It turns out that in most of mathematics, and I never understood quite exactly why this was the case, people don't like square root of 2s in the denominator. Or in general they don't like irrational numbers in the denominator. Irrational numbers are numbers that have decimal places that never repeat and never end. So the way that they get rid of irrational numbers in the denominator is that you do something called rationalizing the denominator. And the way you rationalize a denominator-- let's take our example right now. If we had C over the square root of 2, we just multiply both the numerator and the denominator by the same number, right? Because when you multiply the numerator and the denominator by the same number, that's just like multiplying it by 1. The square root of 2 over the square root of 2 is 1. And as you see, the reason we're doing this is because square root of 2 times square root of 2, what's the square root of 2 times square root of 2? Right, it's 2. Right? We just said, something times something is 2, well the square root of 2 times square root of 2, that's going to be 2. And then the numerator is C times the square root of 2. So notice, C times the square root of 2 over 2 is the same thing as C over the square root of 2. And this is important to realize, because sometimes while you're taking a standardized test or you're doing a test in class, you might get an answer that looks like this, has a square root of 2, or maybe even a square root of 3 or whatever, in the denominator. And you might not see your answer if it's a multiple choice question. What you ned to do in that case is rationalize the denominator. So multiply the numerator and the denominator by square root of 2 and you'll get square root of 2 over 2. But anyway, back to the problem. So what did we learn? This is equal to B, right? So turns out that B is equal to C times the square root of 2 over 2. So let me write that. So we know that A equals B, right? And that equals the square root of 2 over 2 times C. Now you might want to memorize this, though you can always derive it if you use the Pythagorean theorem and remember that the sides that aren't the hypotenuse in a 45-45-90 triangle are equal to each other. But this is very good to know. Because if, say, you're taking the SAT and you need to solve a problem really fast, and if you have this memorized and someone gives you the hypotenuse, you can figure out what are the sides very fast, or i8f someone gives you one of the sides, you can figure out the hypotenuse very fast. Let's try that out. I'm going to erase everything. So we learned just now that A is equal to B is equal to the square root of 2 over 2 times C. So if I were to give you a right triangle, and I were to tell you that this angle is 90 and this angle is 45, and that this side is, let's say this side is 8. I want to figure out what this side is. Well first of all, let's figure out what side is the hypotenuse. Well the hypotenuse is the side opposite the right angle. So we're trying to actually figure out the hypotenuse. Let's call the hypotenuse C. And we also know this is a 45-45-90 triangle, right? Because this angle is 45, so this one also has to be 45, because 45 plus 90 plus 90 is equal to 180. So this is a 45-45-90 triangle, and we know one of the sides-- this side could be A or B-- we know that 8 is equal to the square root of 2 over 2 times C. C is what we're trying to figure out. So if we multiply both sides of this equation by 2 times the square root of 2-- I'm just multiplying it by the inverse of the coefficient on C. Because the square root of 2 cancels out that square root of 2, this 2 cancels out with this 2. We get 2 times 8, 16 over the square root of 2 equals C. Which would be correct, but as I just showed you, people don't like having radicals in the denominator. So we can just say C is equal to 16 over the square root of 2 times the square root of 2 over the square root of 2. So this equals 16 square roots of 2 over 2. Which is the same thing as 8 square roots of 2. So C in this example is 8 square roots of 2. And we also knows, since this is a 45-45-90 triangle, that this side is 8. Hope that makes sense. In the next presentation I'm going to show you a different type of triangle. Actually, I might even start off with a couple more examples of this, because I feel I might have rushed it a bit. But anyway, I'll see you soon in the next presentation.